Abstract

In his biography of Emil Artin, Richard Brauer describes the years from 1931–1941 as a time when “Artin spoke through his students and through the members of his mathematical circle” rather than through written publications. This paper explores these seemingly quiet years when Artin immigrated to America and disseminated ideas about algebraic number theory during this time in his collaboration with George Whaples, a young American mathematician who had just completed his Ph.D. at the University of Wisconsin. The main result of their work is the use of the product formula for valuations to come up with an axiomatic characterization of both algebraic number fields and algebraic function fields with a finite field of constants. These two families of fields are exactly the fields for which class field theory is known to hold. We situate their mathematical work in the broader context of algebraic number theory and their lives within the broader historical context.